The trio Belur, Halebid and Shravanabelagola are three popular excursion destinations from Mysore often done as a circuit tour.

While Belur and Halebid are famous for the Hoysala era temple architecture, Shravanabelagola is an important Jain center.
Mysore to Belur distance is about 160km and Belur to Halebid is another 15km. Shravanabelagola some 50km south of Hassan, the town located in the middle of these three places. You'll need a full day to cover the three places from Mysore.
There are a few options to travel and tour these places from Mysore. Driving to Belur by your car  (or even hiring one), book in a conducted tour from Mysore or travel by the public transport (bus/train) are three viable choices. Select one option depending on your situation.

While the first option (car) is the most flexible for obvious reasons, the last option (bus/train) is highly budget friendly. The conducted tour is a popular via media option between the two.
Let's see the road option first. For the backpackers and budget travelers the public transport options to explore these places are discussed a bit later in this page.

The route from Mysore to Belur ( distance : 154km ) is like this :  Mysore City -- 27 km ----> Bilikere deviation for SH57 --- 13km --->  KR Nagara --- 46km ---> Hole Narasipura  ---33km ---> Hassan --- 35km ---> Belur.
The road is in great condition all the way except the stretch in the outskirts of Hassan.
The first task is to come out of Mysore City and get to the state highway number 88. From the city center take the Hunsur Road (this is SH88) and drive towards the direction of Hunsur (west of Mysore). In about 8km from the city center, you will cross the Outer Ring Road (ORR) junction. Right leads to Bangalore, left go towards Ooty and Calicut.

Drive ahead without any deviations. You are on the 4 lane highway (SH 88). About 7km later Yelwal village arrives and at 21km from the ORR junction mentioned before, you will see the diversion to Hassan on the right. Watch out for this junction ( Bilikere junction ) as it is easy to whiz past it unnoticed. After about 12km from  Yelwal look for that fuel station on the right where the SH88 makes a broad leftward bend.
Now leave the Hunsur Road (SH88) and take the right deviation to Hassan. This is also a state highway (SH 57). Though your road is now 2 lane, the traffic is almost nil compared to the 4lane SH88 you've just left. Nevertheless the road condition is pretty decent for long drives.

You'll cross many small villages and two small towns at regular intervals on SH57 before finally arrive at Hassan. The two small towns you'll cross are KR Nagara and Hole Narasipura. Do not miss the route at the town squares with many deviations. If in doubt , just confirm for direction to Hassan.

From Mysore you should reach Hassan town in about 2 hours. You probably will take a break at Hassan, though you can bypass the town center. This is a large town with many hotels and restaurants. You may not find many such restaurants further in the journey to Belur, the way you've noticed it till Hassan.

At Hassan you'll cross the NH48 ( also Bangalore-Mangalore Road/ BM Road ). From Hassan town, drive a few km towards the Mangalore direction on the NH48 till you see the Belur diversion on your right.
It's again a less busy road from this diversion till you hit Belur. By the way on this road, shortly after the diversion is the popular Hoysala Village Resort. You'll see it on the left.

From Hassan you should reach Belur town in about 1hour time. The temple is located near the Belur bus stand in the town center. Take your car right to the front of the temple where the parking space is located.

Halebid is about 15km east of Belur. There is a good road connection between the two. This is a narrow, winding road - some times through the forest region - with beautiful ancient banyan trees making the canopy. There are ample sign boards in Belur town to give directions to the Halebid road. It is a relatively slow but enjoyable drive between Belur and Halebid.
You need not have to come back to Belur from Halebid to reach Hassan, as there is a different (shorter) route to reach Hassan from Halebid (30km).

If Shravanabelagola is part of your itinerary, head to Channarayapatna (about 38km towards Bangalore direction) from Hassan. A little after Channarayapatna town square, on the NH48 take the right deviation to Shravanabelagola.
The route distances are : Halebid --- 30km ---> Hassan  --- 38km ---> Channarayapatna  --- 13km ---> Shravanabelagola--- 80km --->Mysore city.

By the way there is a great drive in restaurant ( Kamat Upachar) near Channarayapatna on the NH48 on your left. (Planning tip: Those returning to Bangalore can take the NH48, instead of heading back to Mysore )
Depending on the itinerary you may take an altogether different route to Belur (and Halebid) from Mysore. For example if you are planing to visit Shravanabelagola before Belur or Halebid, you would be taking the Pandavapura (via Srirangapatna) route from Mysore. See also the Mysore to Shravanabelagola route.

A tip if you are starting from Mysore in the morning. Leave as early as possible (say about 6am, or even before). You'll reach Belur by about 10am. Complete Belur and Halebid.  After lunch head towards Shravanabelagola. Start back from Shravanabelagola in the evening and return to Mysore by night.
If you do it in the reverse direction, you could reach Shravanabelagola in under 2 hours from Mysore before it is too hot to climb the hill.

So that's it about the road routes, driving directions and distances to Belur, Halebid and Shravanabelagola from Mysore.
A promised, here are the details about the public transport scenario.
There are many direct bus services (KSRTC) to Belur from Mysore. Or you can travel up to Hassan by train and then by local bus to Belur.

Let's see the bus options first. Catch one of those Chikmagalur bound buses that goes via Belur. There are about 6 daily services (Express buses called Karnataka Sarige ) from Mysore to Chikmagalur, most of them leaves Mysore KSRTC Bus stand in the morning (between 6 and 9.30 am) and one in the evening. You can do advance seat reservation (see KSRTC Booking) in one of those services.
These are not the typical upscale services with push back seats, AC etc. The seats are in rows of 2+3 formats. While booking at KSRTC you can opt for the seats you like. Prefer the one in the front rows ( and if you like the window seats!) for obvious reasons.

The distance from Mysore to Belur by bus is 159km that takes about 4 hours by bus. A one way reserved ticket cost Rs116. There are no direct buses to Halebid from Mysore. However you can easily get a local connection to Halebid from Belur and also from Hassan.
For any chance you are not able to catch one of those direct buses to Belur from Mysore, look for a connection to Hassan from Mysore.In addition to bus options, you can also explore the possibility of a train connection.
There are many daily and weekly trains from Mysore that pass via Hassan.
Shimoga Expres (Train No: 16206) leaves by 6 am ( Rs76 second class  ) while the Dharwad Express (Train No: 17301) and Kannur Express (Train No :16517) are late night trains from Mysore. Sharavati Express (Train No:11036, on Thursdays) and Swarna Jayanthi Express ( Train No: 12781 , on Fridays) leaves in the mornings from Mysore. Check the IRCTC website for schedules and reservations.

What about the organized tours, third option?
There are many private and state owned (KSTDC) agencies operate day trips to the Belur, Halebid and Shravanabelagola circuit. The frequencies and types of buses vary largely with season. They start from Mysore city. In general they all leave early in the morning (6'ish) and drop you back in the city by the dinner time.
The advantages are may. You are free from the troubles of arranging your own transport to all these places. This is a conducted tour, so they take care of the schedules and drop you back in the city by night. You are probably tired but it was hassle free.
The major disadvantage of these day trips is of course its packed schedule. Traveling up and down, about 8 hours you will be in the bus. You may spend about an hour each at these attractions, which is a bit too less for these sites. So ultimately its a compromise in the schedules and cost, when you do it as a conducted day trip.
The tariff for the package is broadly around Rs1000 per head with variations depending on the operator, season and type of bus etc. Of all the KSTDC's is the most popular. Mostly the tariffs quoted doesn't include the food, entry ticket etc though they may provide a guide at the site. Ask your hotel travel desk for assistance and suggestion.
Tailpiece: Consider this itinerary idea if it is of help. Knock one day off from your Mysore agenda.  Book your inward and outward travel from Hassan (by bus or train) preceding or following the Mysore itinerary. At Hassan arrange for a car for the sightseeing. This means staying at Hassan for a night before you arrive at Mysore or as you leave after the Mysore tour. Keeping Hassan as base for  Belur, Halebid and Shravanabelagola circuit  is a lot relaxing from the schedule point. You'll not regret that you've done that!